scholarly journals On the regularity of timelike extremal surfaces

2014 ◽  
Vol 17 (01) ◽  
pp. 1450048 ◽  
Author(s):  
Robert L. Jerrard ◽  
Matteo Novaga ◽  
Giandomenico Orlandi
Keyword(s):  
Hausdorff Dimension ◽  
Minkowski Space ◽  
Specific Form ◽  
Singular Set ◽  
Natural Topology ◽  
One Dimensional ◽  
Class C

We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+n, characterized by the fact that they admit a C1 parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class Ck, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1 + 1/k, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in ℝ1+n is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

2009 ◽  
Vol 19 (02) ◽  
pp. 545-555 ◽  
Author(s):  
F. TRAMONTANA ◽  
L. GARDINI ◽  
D. FOURNIER-PRUNARET ◽  
P. CHARGE
Keyword(s):  
Focal Point ◽  
Singular Line ◽  
Two Dimensional ◽  
Focal Points ◽  
Singular Set ◽  
One Dimensional ◽  
Attracting Set ◽  
Straight Line ◽  
Special Character ◽  
Stable And Unstable Sets

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

scholarly journals Regularity of area minimizing currents mod p

2020 ◽  
Vol 30 (5) ◽  
pp. 1224-1336
Author(s):  
Camillo De Lellis ◽  
Jonas Hirsch ◽  
Andrea Marchese ◽  
Salvatore Stuvard
Keyword(s):  
Hausdorff Dimension ◽  
Partial Regularity ◽  
Singular Set ◽  
Regularity Theorem ◽  
Locally Finite ◽  
Dimensional Measure ◽  
Area Minimizing ◽  
Mod P

AbstractWe establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current $${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than $$m-1$$ m - 1 . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ ( m - 1 ) -rectifiable with locally finite $$(m-1)$$ ( m - 1 ) -dimensional measure.

A measureless one-dimensional set

1954 ◽  
Vol 50 (3) ◽  
pp. 391-393
Author(s):  
H. G. Eggleston
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Single Point ◽  
Outer Measure ◽  
One Dimensional ◽  
Infinite Measure

It has been known for some time that there are sets in Euclidean space which are of infinite measure in a certain Hausdorff dimension, say α, and yet which contain no subsets that are of finite positive a measure. The properties of such a set, say X, could be developed in much the same way as those of sets which are of positive finite α measure, if there existed a Caratheodory outer measure Γ, defined over the subsets of X, which was such that ∞ > Γ(X) > 0, and for any subset Y of X, Λα (Y) = 0 implied Γ (Y) = O. The object of this note is to show that there are sets for which no such outer measure exists. It is shown that a set defined by Sierpinski (7) is one-dimensional and is such that any Caratheodory outer measure defined over its subsets either takes infinite values or is identically zero or is not zero for all subsets that consist of a single point. It has been remarked by Prof. Besicovitch that these properties imply that the set is measurable with respect to any Caratheodory outer measure defined over subsets of the plane, and in fact any subset is measurable with respect to any such outer measure.

ABSORPTION OF OPTICAL PULSES UNDER PROPAGATION THROUGH ONE-DIMENSIONAL RESONANT FRACTAL CLUSTERS

Fractals ◽  
1994 ◽  
Vol 02 (04) ◽  
pp. 553-556 ◽  
Author(s):  
YU. V. KISTENEV ◽  
A. V. SHAPOVALOV
Keyword(s):  
Numerical Simulation ◽  
Hausdorff Dimension ◽  
Dimensional Structure ◽  
Optical Pulses ◽  
One Dimensional ◽  
Fractal Object ◽  
Fractal Clusters ◽  
Fractal Media

Peculiarities of propagation of short optical pulses through resonantly absorbing fractal media are investigated in the work using methods of numerical simulation. The fractal object was modeled by a one-dimensional structure with its parameters being similar to generalized Cantor dust. The dependence of transmission of such a medium on its Hausdorff dimension in some cases was shown to have a nonmonotonous character. Interpretation of the results obtained is also given.

Unfolding plane curves with cusps and nodes

2015 ◽  
Vol 145 (1) ◽  
pp. 161-174
Author(s):  
Juan J. Nuño Ballesteros
Keyword(s):  
Finite Codimension ◽  
Plane Curves ◽  
Singular Set ◽  
One Dimensional ◽  
Double Points ◽  
Euler Obstruction ◽  
Type Singularity ◽  
Transverse Slice

Given an irreducible surface germ (X, 0) ⊂ (ℂ3, 0) with a one-dimensional singular set Σ, we denote by δ1 (X, 0) the delta invariant of a transverse slice. We show that δ1 (X, 0) ≥ m0 (Σ, 0), with equality if and only if (X, 0) admits a corank 1 parametrization f :(ℂ2, 0) → (ℂ3, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(X, 0) in order to characterize those surfaces that have finite codimension with respect to -equivalence or as a frontal-type singularity.

scholarly journals MATHEMATICAL MODELS FOR COMPUTERIZED CONTROL SYSTEM

2019 ◽  
Vol 48 (05) ◽  
pp. 119-123
Author(s):  
Safwan Al Salaimeh
Keyword(s):  
Control System ◽  
Mathematical Models ◽  
Control Systems ◽  
Control Function ◽  
Specific Form ◽  
Mathematical Methods ◽  
One Dimensional ◽  
Systems Models ◽  
Nonlinear Static ◽  
Lumped Parameters

The software is a set of mathematical methods, and algorithms of information processing, which used in creating the control system. When designing control systems, Initial data for the design of control system. The tasks of the computerized control system are understood as a part of the computerized functions of the computerized control system characterized by the outcomes and outputs in specific form. control function is: commutative action for computerized control system, aimed to achieve a criterion goal. Depending on the properties of the process and their mathematical description can be combined into different classes; This paper shows the designing the mathematical models which need to computerized control systems (models (3) – (8)). In the same time this paper shows the main methods which were used to formulate the mathematical models as: • Stochastic and deterministic; • One dimensional and multidimensional; • Linear and nonlinear; • Static and dynamic; • Stationary and non – stationary; • With distributed and lumped parameters.

Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

2001 ◽  
Vol 44 (2) ◽  
pp. 150-159 ◽  
Author(s):  
Piotr Jakóbczak
Keyword(s):  
Unit Ball ◽  
Lebesgue Measure ◽  
Bergman Space ◽  
Dimensional Subspace ◽  
Natural Topology ◽  
Exceptional Set ◽  
One Dimensional ◽  
Exceptional Sets

AbstractLet BN be the unit ball in and let f be a function holomorphic and L2-integrable in BN. Denote by E(BN, f) the set of all slices of the form , where L is a complex one-dimensional subspace of , for which is not L2-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for f. We give a characterization of exceptional sets which are closed in the natural topology of slices.

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